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Quasiconvexity is a more general property than convexity in that all convex functions are also quasiconvex, but not all quasiconvex functions are convex. Univariate unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments .
Quasiconvexity is a generalisation of convexity for functions defined on matrices, to see this let and ((,),) with (,) =.The Riesz-Markov-Kakutani representation theorem states that the dual space of () can be identified with the space of signed, finite Radon measures on it.
If takes only finite values, then polyconvexity implies quasiconvexity and thus leads to the weak lower semicontinuity of the corresponding integral functional on a Sobolev space. If m = 1 {\displaystyle m=1} or n = 1 {\displaystyle n=1} , then polyconvexity reduces to convexity.
Quasiconvexity of the Lagrangian density, Morrey proved, is equivalent to the lower-semicontinuity of the action. It just seems natural to include this somewhere. A good reference is Bernard Dacorogna's book, Direct Methods in the Calculus of Variations, part II.
A fractional program in which f is nonnegative and concave, g is positive and convex, and S is a convex set is called a concave fractional program.If g is affine, f does not have to be restricted in sign.
Quasiconvexity (calculus of variations) R. Regularized canonical correlation analysis; S. Saint-Venant's theorem; Signorini problem; Stampacchia Medal; T.
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The direct method may often be applied with success when the space is a subset of a separable reflexive Banach space.In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence () in has a subsequence that converges to some in with respect to the weak topology.